Also, it is known (see, e.g., [1]}, [2]}) that \(U(x)\)  is the unique (up to translations) extremal of the following Gagliardo-Nirenberg type inequality
\(\Vert u\Vert _{\frac{4s}{N}+2}^{\frac{4s}{N}+2}\le \frac{N+2s}{Na_s^{*}}\Vert (-\Delta )^{\frac{s}{2}}u\Vert _{2}^{2}\Vert u\Vert _{2}^{\frac{4s}{N}},\ \forall u\in H^{s}(\mathbb {R}^{N}),\) 
